Problem: Simplify and expand the following expression: $ \dfrac{a}{5a + 7}-\dfrac{a + 1}{a - 7} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5a + 7)(a - 7)$ Multiply the first term by $\dfrac{a - 7}{a - 7}$ $ \begin{align*} \dfrac{a}{5a + 7} \times \dfrac{a - 7}{a - 7} & = \dfrac{(a)(a - 7)}{(5a + 7)(a - 7)} \\ & = \dfrac{a^2 - 7a}{(5a + 7)(a - 7)}\end{align*} $ Multiply the second term by $\dfrac{5a + 7}{5a + 7}$ $ \begin{align*} \dfrac{a + 1}{a - 7} \times \dfrac{5a + 7}{5a + 7} & = \dfrac{(a + 1)(5a + 7)}{(a - 7)(5a + 7)} \\ & = \dfrac{5a^2 + 12a + 7}{(a - 7)(5a + 7)}\end{align*} $ Now we have: $ = \dfrac{a^2 - 7a}{(5a + 7)(a - 7)} - \dfrac{5a^2 + 12a + 7}{(a - 7)(5a + 7)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{a^2 - 7a - (5a^2 + 12a + 7)}{(5a + 7)(a - 7)} $ $ = \dfrac{a^2 - 7a - 5a^2 - 12a - 7}{(5a + 7)(a - 7)} $ $ = \dfrac{-4a^2 - 19a - 7}{(5a + 7)(a - 7)}$ Expand the denominator: $ = \dfrac{-4a^2 - 19a - 7}{5a^2 - 28a - 49}$